The posterior inconsistency of the Bayesian fused lasso method motivates us to explore other approaches that would mitigate the problems leading to the undesirable behavior of the posterior distribution. Shrinkage priors qualify as naturally good choices for our problem, as they can address the dual issue of shrinking true zero parameters to zero, and retrieving the `boundaries' of the blocks effectively by placing sufficient mass on the non-zero values of successive differences of the normal means. In the normal sequence model, optimal posterior concentration has been achieved via using shrinkage prior distributions with polynomially decaying tails [1]}. This is in contrast to the Laplace prior, that has exponentially light tails.
